Wallis' product exploration

[Allen]

maybe a more compact way of looking at the prime powers in the denominator:

here \( 08002222022000022222222221 \rightarrow 2^{0} 3^{8} 5^{0} 7^{0} 11^{2} 13^{2} 17^{2} 19^{2} 23^{0} 29^{2} 31^{2} 37^{0} 41^{0} 43^{0} 47^{0} 53^{2} 59^{2} 61^{2} 67^{2} 71^{2} 73^{2} 79^{2} 83^{2} 89^{2} 97^{2} 101^{1} \)

The carats under each line indicate the prime factors of n... and a * next to n represents prime n.

Code:

 
  n   prime exponents in denominator from p=3..101]
 50 08002222022000022222222221
    ^ ^                       
 49 06401222022000022222222220
       ^                      
 48 04440222022000022222222210
    ^^                        
*47 06340221022000022222222200
                  ^           
 46 05240220021000222222222200
    ^       ^                 
 45 04230120220000222222222200
     ^^                       
 44 08420020220000222222222100
    ^   ^                     
*43 07422020210000222222222000
                 ^            
 42 06322010200002222222222000
    ^^ ^                      
*41 08242000200002222222221000
                ^             
 40 04242000200022222222220000
    ^ ^                       
 39 00442000200022222222210000
     ^   ^                    
 38 02431200200022222222200000
    ^      ^                  
*37 01220202200022222222200000
               ^              
 36 00020202200222222222100000
    ^^                        
 35 04020202200222222221000000
      ^^                      
 34 03240202100222222220000000
    ^     ^                   
 33 02240222000222222210000000
     ^  ^                     
 32 04142122000222222200000000
    ^                         
*31 02032022000222222200000000
              ^               
 30 00022022002222222100000000
    ^^^                       
*29 02222022002222221000000000
             ^                
 28 01222021022222220000000000
    ^  ^                      
 27 00141020022222220000000000
     ^                        
 26 06040020022222210000000000
    ^    ^                    
 25 05040210022222200000000000
      ^                      
 24 04420200022222200000000000
    ^^                      
*23 06400200022222100000000000
            ^               
 22 04300200222222000000000000
    ^   ^                  
 21 02202200222221000000000000
     ^ ^                   
 20 04222200222210000000000000
    ^ ^                   
*19 03422100222200000000000000
           ^             
 18 02422002222100000000000000
    ^^                  
*17 06312002222000000000000000
          ^            
 16 05201022222000000000000000
    ^                 
 15 04200022221000000000000000
     ^^              
 14 06400022210000000000000000
    ^  ^            
*13 03420022200000000000000000
         ^         
 12 00220222200000000000000000
    ^^             
*11 02020222100000000000000000
        ^         
 10 01012222000000000000000000
    ^ ^         
  9 00202221000000000000000000
     ^         
  8 04202210000000000000000000
    ^          
 *7 03102200000000000000000000
       ^     
  6 02022100000000000000000000
    ^^       
 *5 04021000000000000000000000
      ^     
  4 02220000000000000000000000
    ^     
 *3 00210000000000000000000000
     ^   
 *2 02100000000000000000000000
    ^   
  1 01000000000000000000000000